多自由度杜芬微分方程組的保能及保群計算方法

Abstract

在工程與數學應用中，非線性振動是個相當常見的問題，過去的文獻中已經有許多方法可以求解非線性振動的問題，但它們往往都忽略了能量守恆這個議題。在本篇論文中，我們提出了保能算法(EPS)，將無阻尼與無外力情況下的杜芬微分方程組透過李群轉換成常微分方程組，求解過程中保能算法能夠自動保持能量守恆，使得能量在長時間計算維持不變；接下來我們會繼續探討加上阻尼與外力的杜芬微分方程組，這個部分我們將使用有效且具有高精確度的保群算法(GPS)求解。最後，我們會將整個問題延伸到二維空間與三維空間中，同樣地，我們可以使用保能算法與保群算法去求解二維與三維的杜芬微分方程組。此外，我們將使用四階龍格－庫塔(RK4)方法與保能算法以及保群算法做比較，因為四階龍格－庫塔方法能夠有效地求解微分方程問題，且同時具有四階的精度，所以求出的結果相當值得我們信賴。藉由與四階龍格－庫塔方法求得之解做比較，我們可以得知EPS與GPS的優點、精確性，當然還可以藉由比較每一步所產生的能量誤差得知EPS的保能效果與優越性。

In engineering and mathematics fields, the oscillatory problems of nonlinear oscillators are common problems. There are many computational methods which have been developed for solving the nonlinear oscillatory problems. However, most of these methods can not retain the energy. In this thesis, we develop a novel energy preserving scheme (EPS) for the undamped and unforced Duffing equation by recasting it to a Lie-type ordinary differential equation. The EPS can automatically preserve the total energy to be a constant value in a long term computation. Then, we will extend this problem to the damped and forced Duffing equations. Here, we use the group preserving schemes (GPS) to solve the problems, which can solve the problems effectively and accurately. Finally, we extend the problems to the coupled Duffing equations and three degrees of freedom Duffing equations. Also, we still can use the EPS and the GPS to solve the problems accurately. In each problem, we also compare the present results with the solution obtained by the fourth order Runge-Kutta (RK4) method, which has fourth-order accuracy. By comparing the EPS and RK4, we can see the advantages, accuracy and capability of preserving energy of the EPS.